DWAI and UPM are violations and not crimes whereas DWI is a crime in in New York and if convicted of it you do have a criminal record. I cannot imagine that there is anything illegal with simply submitting an application so I don't believe that charges could be brought (unless of course you lie on the application. You should post this question in Utah and/or Florida because it is their rules that you are looking for clarification on.
I have seen the Utah and Florida applications. I believe there is a question that specifically asks whether you have ever been disqualified from receiving a license to carry a pistol from any municipality, and another that asks whether or not you have ever had a license to carry revoked. Obviously you have to answer these questions truthfully, otherwise you could face criminal charges (since the applications require you to swear to the truth of its content).
I'm not sure a NY lawyer is hte best person to ask. I suggest you post this on the Florida and Utah Avvo pages.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 385-8015 or via email at Eric@RothsteinLawNY.com. The above answer is for informational purposes only and not meant as legal advice.